Linear systems x’ = Ax.

$$B = \left( {{{b}_{{ij}}}} \right) = \left( {\begin{array}{*{20}{c}} {{{b}_{{11}}}\quad {{b}_{{12}}}\quad \cdots \quad {{b}_{{1s}}}} \hfill \\ {{{b}_{{21}}}\quad {{b}_{{22}}}\quad \cdots \quad {{b}_{{2s}}}} \hfill \\ \vdots \hfill \\ {{{b}_{{r1}}}\quad {{b}_{{r2}}}\quad \cdots \quad {{b}_{{rs}}}} \hfill \\ \end{array} } \right) = \left( {{{b}^{1}}{{b}^{2}} \ldots {{b}^{s}}} \right)$$ is an r×s matrix (real or complex),where $${{b}^{j}} = \left( {\begin{array}{*{20}{c}} {{{b}_{{1j}}}} \\ {{{b}_{{2j}}}} \\ \vdots \\ {{{b}_{{rj}}}} \\ \end{array} } \right)$$ is the $$j\mathop {th}\limits_ - - colum$$ vector in B.Thus,for $$c = \left( {{{c}_{i}}} \right) = \left( {\begin{array}{*{20}{c}} {{{c}_{1}}} \\ \vdots \\ {{{c}_{s}}} \\ \end{array} } \right)$$ any s-vector $$Bc = {c_1}{b^1} + {c_2}{b^2} + \cdots + {c_s}{b^s}$$ .